When showing that the policy improvement theorem applies to MC control, why is $q_{\pi_{k}}\left(s, \pi_{k}(s)\right) \geq v_{\pi_{k}}(s)$ true?
You are right that the strict equality $q_\pi(s,\pi(s)) = v_\pi(s)$ is generally true for a deterministic policy $\pi$.
The $\geq$ inequality is also correct, of course, and it could be that the authors' intention was to show that $\pi_{k+1}$ and $\pi_k$ satisfy the condition for the policy improvement theorem:

Let $\pi$ and $\pi'$ be any pair of deterministic policies such that, for all $s\in\mathcal{S}$,
$$q_\pi(s,\pi'(s))\geq v_\pi(s)\tag{4.7}$$

So they kinda relaxed the equality to show that the condition is satisfied &quot;word-for-word&quot; for $\pi'=\pi_{k+1}$ and $\pi=\pi_k$.
Although, I must say that they don't seem to be consistent in this - for example, they keep the strict equality at the derivation at the bottom of the page 78. So it might as well be that you've discovered a typo in Sutton and Barto.